“Romney wants to, he said in the first 100 days, he’s going to let the big banks again write their own rules, unchain Wall Street,” Mr. Biden said at a campaign rally in Danville, Va.

The vice president then lowered his voice and said, “They’re going to put y’all back in chains.”

Some in the audience laughed, but Republicans said the Obama campaign lowered itself to new depths of gutter politics.

“In case anyone was wondering just how low President Obama could go in his campaign for re-election, we now know he’s willing to say that Governor Romney wants to put people back in chains,” Romney campaign spokeswoman Andrea Saul said in a statement. “Whether it’s accusing Mitt Romney of being a felon, having been responsible for a woman’s tragic death or now wanting to put people in chains, there’s no question that because of the president’s failed record he’s been reduced to a desperate campaign based on division and demonization.”

Ari Fleischer, a White House press secretary for George W. Bush, said Mr. Biden’s remark was “objectionable.”

“The press pounded [Sarah] Palin when she talked about ‘blood libel,’” Mr. Fleischer said via Twitter. “What will they do about Biden’s ‘chains’ remark?”

The Romney campaign demanded to know what Mr. Obama, the nation’s first black president, thought about his running mate’s racially charged comment. Within hours, an Obama campaign official said the president was fine with it.

The Democrats’ camp said Republicans such as House Speaker John A. Boehner of Ohio and Rep. Paul Ryan of Wisconsin — Mr. Romney’s running mate — had used similar rhetoric.

“For months, Speaker Boehner, Congressman Ryan and other Republicans have called for the ‘unshackling’ of the private sector from regulations that protect Americans from risky financial deals and other reckless behavior that crashed our economy,” said Obama deputy campaign manager Stephanie Cutter.

“Since then, the vice president has often used a similar metaphor to describe the need to ‘unshackle’ the middle class. Today’s comments were a derivative of those remarks, describing the devastating impact letting Wall Street write its own rules again would have on middle-class families,” she said.

A review of transcripts of public officials’ speeches and news conferences since January showed that only one Republican leader, House Majority Whip Kevin McCarthy of Texas, has used similar language, once on March 27 and again on Aug. 1.

Speaking about the need to reduce regulations to help the economy, Mr. McCarthy said, “We believe [the economy] can be stronger if you unshackle what holds us back.”

The Obama team provided no examples where the speech was to a largely black audience, used the second-person “you” as an address or for emphasis, or talked about putting people “back in chains.”

Later, at a campaign event in Wytheville, Va., Mr. Biden changed his words, saying “in shackles” rather than “in chains” and noting that House Republicans have talked about plans to “unshackle” the economy.

“I am told,” Mr. Biden said, “when I made that comment earlier today in Danville, Va., the Romney campaign put out a tweet, put out a tweet, went on the air, went on the airwaves saying, ‘Biden was outrageous in saying’ — I think I said, instead of unshackled, unchained — ‘outrageous to say that!’”

He added: “I got a message for them: If you want to know what’s outrageous, it’s their policies. Their effects on middle-class Americans, that’s what’s outrageous. That’s what’s outrageous.”

Mr. Biden’s comments are the latest salvo in a campaign that has grown increasingly nasty.

Ms. Cutter suggested several months ago that Mr. Romney may have committed a felony by misrepresenting his position at Bain Capital in documents filed with the Securities and Exchange Commission.

Senate Majority Leader Harry Reid, Nevada Democrat, has said that a person, whom he did not identify, told him that Mr. Romney hasn’t paid taxes at all for 10 years. Mr. Reid didn’t offer proof and said it was unnecessary.

Last week, a pro-Obama super PAC released an advertisement that essentially blames Mr. Romney for the death of a steelworker’s wife who developed cancer five years after Bain Capital closed a steel plant. Although the super PAC is an independent actor, several Obama campaign officials who have been offered chances to disassociate their team from the ad have not criticized it or gone significantly beyond saying it isn’t their ad.

Democrats also counter that pro-Romney advocacy groups have run ads still questioning whether Mr. Obama is an American citizen.

For example, an organization known as the Conservative Majority Fund has run a national ad campaign claiming that it can disqualify Mr. Obama from running for re-election.

“No one has seen an actual physical copy of Barack Obama’s birth certificate,” states the ad, which has appeared on CNN and Fox. “The fact is, if we don’t know who Barack Obama is, we shouldn’t even have him as a candidate for president.”